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Supplementary Notes
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Present Value
Net Present Value
NPV Rule
Opportunity Cost of Capital
1
Present Value
Present Value
Discount Factor
Value today of a
future cash
flow.
Present value of
a $1 future
payment.
Discount Rate
Interest rate used
to compute
present values of
future cash flows.
2
Present Value
Present Value = PV
PV = discount factor  C1
3
Present Value
Discount Factor = DF = PV of $1
DF 
1
(1 r ) t
Discount Factors can be used to compute the present value of
any cash flow.
4
Valuing an Office Building
Step 1: Forecast cash flows
Cost of building = C0 = 350
Sale price in Year 1 = C1 = 400
Step 2: Estimate opportunity cost of capital
If equally risky investments in the capital market
offer a return of 7%, then
Cost of capital = r = 7%
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Valuing an Office Building
Step 3: Discount future cash flows
PV 
C1
(1r )

400
(1.07)
 374
Step 4: Go ahead if PV of payoff exceeds
investment
NPV  350  374  24
6
Net Present Value
NPV = PV - required investment
C1
NPV = C0 
1 r
7
Risk and Present Value
• Higher risk projects require a higher rate of
return.
• Higher required rates of return cause lower PVs.
PV of C1  $400 at 7%
400
PV 
 374
1  .07
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Risk and Present Value
PV of C1  $400 at 12%
400
PV 
 357
1  .12
PV of C1  $400 at 7%
400
PV 
 374
1  .07
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Net Present Value Rule
• Accept investments that have positive net
present value.
Example
Suppose we can invest $50 today and
receive $60 in one year. Should we accept
the project given a 10% expected return?
60
NPV = -50 +
 $4.55
1.10
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Opportunity Cost of Capital
Example
You may invest $100,000 today. Depending on
the state of the economy, you may get one of
three possible cash payoffs:
Economy
Payoff
Slump
Normal Boom
$80,000 110,000 140,000
80,000  110,000  140,000
Expected payoff  C1 
 $110,000
3
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Opportunity Cost of Capital
Example - continued
The stock is trading for $95.65. Depending on
the state of the economy, the value of the stock
at the end of the year is one of three
possibilities:
Economy Slump
Stock Pric e $80
Normal
110
Boom
140
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Opportunity Cost of Capital
Example - continued
The stocks expected payoff leads to an
expected return.
80  100  140
Expected payoff  C1 
 $110
3
expected profit 110  95.65
Expected return 

 .15 or 15%
investment
95.65
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Opportunity Cost of Capital
Example - continued
Discounting the expected payoff at the expected
return leads to the PV of the project.
110,000
PV 
 $95,650
1.15
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Investment vs. Consumption
• Some people prefer to consume now.
Some prefer to invest now and consume
later.
• Borrowing and lending allows us to
reconcile these opposing desires which
may exist within the firm’s shareholders.
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Investment vs. Consumption
income in period 1
100
An
80
Some investors will prefer A
and others B
60
40
Bn
20
20
40
60
income in period 0
80
100
16
Investment vs. Consumption
The grasshopper (G) wants to
consume now. The ant (A) wants to
wait. But each is happy to invest. A
prefers to invest 14%, moving up
the red arrow, rather than at the 7%
interest rate. G invests and then
borrows at 7%, thereby
transforming $100 into $106.54 of
immediate consumption. Because
of the investment, G has $114 next
year to pay off the loan. The
investment’s NPV is $106.54-100 =
17
+6.54
Investment vs. Consumption
•
Dollars
Later
A invests $100 now
and consumes $114
next year
114
107
The grasshopper (G) wants to consume
now. The ant (A) wants to wait. But each
is happy to invest. A prefers to invest
14%, moving up the red arrow, rather than
at the 7% interest rate. G invests and
then borrows at 7%, thereby transforming
$100 into $106.54 of immediate
consumption. Because of the investment,
G has $114 next year to pay off the loan.
The investment’s NPV is $106.54-100 =
+6.54
G invests $100 now,
borrows $106.54 and
consumes now.
100
106.54
Dollars
Now
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Topics Covered
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Valuing Long-Lived Assets
PV Calculation Short Cuts
Compound Interest
Interest Rates and Inflation
Example: Present Values and Bonds
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Present Values
Discount Factor = DF = PV of $1
DF 
1
t
(1 r )
• Discount Factors can be used to
compute the present value of any cash
flow.
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Present Values
C1
PV  DF  C1 
1  r1
DF 
1
(1 r ) t
• Discount Factors can be used to
compute the present value of any cash
flow.
21
Present Values
Ct
PV  DF  Ct 
1  rt
• Replacing “1” with “t” allows the formula
to be used for cash flows that exist at
any point in time.
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Present Values
Example
You just bought a new computer for $3,000. The
payment terms are 2 years same as cash. If you can
earn 8% on your money, how much money should
you set aside today in order to make the payment
when due in two years?
PV 
3000
(1.08) 2
 $2,572.02
23
Present Values
• PVs can be added together to evaluate
multiple-periods cash flows.
PV 
C1
(1 r )
 (1r ) 2 ....
C2
1
24
Present Values
• Given two dollars, one received a year from now
and the other two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20% and r2 = 7%.
25
Present Values
• Given two dollars, one received a year from now
and the other two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20% and r2 = 7%.
DF1 
1.00
(1.20)1
 .83
DF2 
1.00
(1.07 ) 2
 .87
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Present Values
Example
Assume that the cash flows
from the construction and
sale of an office building is
as follows. Given a 7%
required rate of return,
create a present value
worksheet and show the net
present value.
Year 0
Year 1
Year 2
 150,000  100,000  300,000
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Present Values
Example - continued
Assume that the cash flows from the construction and sale of an
office building is as follows. Given a 7% required rate of return,
create a present value worksheet and show the net present
value.
Period
0
1
2
Discount
Factor
1.0
1
1.07  .935
1
 .873
1.07 2
Cash
Present
Flow
Value
 150,000
 150,000
 100,000
 93,500
 300,000
 261,900
NPV  Total  $18,400
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Short Cuts
• Sometimes there are shortcuts that make
it very easy to calculate the present value
of an asset that pays off in different
periods. These tolls allow us to cut
through the calculations quickly.
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Short Cuts
Perpetuity - Financial concept in which a
cash flow is theoretically received forever.
cash flow
Return 
present va lue
C
r
PV
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Short Cuts
Perpetuity - Financial concept in which a
cash flow is theoretically received forever.
cash flow
PV of Cash Flow 
discount rate
C1
PV 
r
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Short Cuts
Annuity - An asset that pays a fixed sum
each year for a specified number of years.
1
1 
PV of annuity  C   
t
 r r 1  r  
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Annuity Short Cut
Example
You agree to lease a car for 4 years at $300 per
month. You are not required to pay any money up
front or at the end of your agreement. If your
opportunity cost of capital is 0.5% per month, what is
the cost of the lease?
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Annuity Short Cut
Example - continued
You agree to lease a car for 4 years at $300
per month. You are not required to pay any
money up front or at the end of your
agreement. If your opportunity cost of
capital is 0.5% per month, what is the cost of
the lease?
 1

1
Lease Cost  300  

48 
 .005 .0051  .005 
Cost  $12,774.10
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Compound Interest
i
ii
Periods Interest
per
per
year
period
iii
APR
(i x ii)
iv
Value
after
one year
v
Annually
compounded
interest rate
1
6%
6%
1.06
6.000%
2
3
6
1.032
= 1.0609
6.090
4
1.5
6
1.0154 = 1.06136
6.136
12
.5
6
1.00512 = 1.06168
6.168
52
.1154
6
1.00115452 = 1.06180
6.180
365
.0164
6
1.000164365 = 1.06183
6.183
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18
16
14
12
10
8
6
4
2
0
10% Simple
30
27
24
21
18
15
12
9
6
10% Compound
3
0
FV of $1
Compound Interest
Number of Years
36